DIGITAL

FILTERS

Presented by Mr.Nilesh Rambahadur Jaiswar Mr.Kunal Dilip Vartak

INTRODUCTION

In signal processing, the function of a filter is to remove unwanted parts of the signal, such as random noise, or to extract useful parts of the signal, such as the components lying within a certain frequency range.

Basic Setup Of Digital Filter

Operation Of Digital Filters :• Raw Signal : V = x(t) Where t is time.• This signal is sampled at time intervals h (the

sampling interval). The sampled value at time t = ih is xi=x(ih)

• Transferred From The ADC To The Processor x0 , x1 , x2 , x3 , ...

• Corresponding to the values of the signal waveform att = 0, h, 2h, 3h, ...

• And t = 0 is the instant at which sampling begins.• At time t = nh the values available to the processor

x0 , x1 , x2 , x3 , ... Xn

• The digital output from the processor to the DACy0 , y1 , y2 , y3 , ... yn

Examples Of Simple Digital Filters Unity Gain Filter:yn = xn

Each output value yn is exactly the same as the corresponding input value xn: y0 = x0 , y1 = x1 , .................

Simple Gain Filter:yn = Kxn

where K = constant.This simply applies a gain factor K to each input

value.

Pure Delay Filter:yn = xn-1

The output value at time t = nh is simply the input at time

t = (n-1)h , i.e. the signal is delayed by time h : y0 = x-1 ,y1 = x0,.....

Two-term Difference Filter:yn = xn– xn-1

The output value at t = nh is equal to the difference between the current input xn and the previous input xn-1 :

y0 = x0– x-1 ,

y1 = x1– x0 ,

y2 = x2– x1 , ... etc

Two-term Average Filter:yn = ( xn + xn-1 )/2The output is the average (arithmetic mean) of the

current and previous input:y0 =(x0 + x-1)/2 , y1 =(x1 + x0)/2 , y2 =(x2 + x1)/2 , ... etc

Three-term Average Filter:yn =(xn + xn-1 + xn-2)/ 3This is similar to the previous example, with the

average being taken of the current and two previous inputs:

y0 =(x0 + x-1 + x-2)/ 3y1 =(x1 + x0 + x-1)/ 3y2 =(x2 + x1 + x0)/ 3As before, x-1 and x-2 are taken to be zero.

Central Difference Filter:yn =(xn– xn-2)/ 2The output is equal to half the change in the input

signal over the previous two sampling intervals:y0 =(x0 – x-2)/ 2y1 =(x1 – x-1)/ 2 , etc….

Types of Digital FiltersFinite Impulse Response , Or FIR Filters :This filters express each output sample as a

weighted sum of the last N input samples, where N is the order of the filter. FIR filters are normally non-recursive, meaning they do not use feedback and as such are inherently stable.

Infinite Impulse Response , Or IIR Filters:This filters are the digital counter part to

analog filters. Such a filter contains internal state, and the output and the next internal state are determined by a linear combination of the previous inputs and outputs

Finite impulse responseIn signal processing , a finite impulse response

(FIR) filter is a filter whose impulse response (or response to any finite length input) is of finite duration, because it settles to zero in finite time.

Definition

PropertiesAre Inherently Stable

This is due to the fact that, because there is no required feedback, all the poles are located at the origin and thus are located within the unit circle .

Require No Feedback:This means that any rounding errors are not

compounded by summed iterations. The same relative error occurs in each

calculation. This also makes implementation simpler .

They Can Easily Be Designed to be linear phase by making the coefficient sequence symmetric; linear phase, or phase change proportional to frequency, corresponds to equal delay at all frequencies.

Impulse responseThe impulse response h[n]can be calculated if we set

x[n]= δ[n]in the above relation, where δ[n] is the Kronecker delta impulse. The impulse response for an FIR filter then becomes the set of coefficients , as follows

For n=0 to N

The Z-transform of the impulse responseFIR filters are clearly bounded - input bounded-output (BIBO) stable, since the output is a sum of a finite number of finite multiples of the input values, so can be no greater than

Filter designTo design a filter means to select the

coefficients such that the system has specific characteristics.

Window design methodIn the Window Design Method, one designs an ideal IIR filter, then applies a window function to it –Ve in the time domain, multiplying the infinite impulse by the window function. This results in the frequency response of the IIR being convolved with the frequency response of the window function. If the ideal response is sufficiently simple, such as rectangular, the result of the convolution can be relatively easy to determine.

Frequency Sampling methodWeighted least squares design

Moving Average Example

Block diagram of a simple FIR filter (2nd-order/3-tap filter in this case, implementing a moving average)

Pole-Zero Diagram

Amplitude And Phase Responses

A moving average filter is a very simple FIR filter. It is sometimes called a boxcar filter.

The filter coefficients, b0,......,bN are found via the following equation:

To provide a more specific example, we select the filter order: N = 2

The impulse response of the resulting filter is:

z-transform of the impulse response:

Infinite Impulse ResponseInfinite Impulse Response (IIR) is a property of signal processing systems. Systems with this property are known as IIR systems or, when dealing with filter systems, as IIR filters.

Implementation And Design

IIR filters may be implemented as either analog or digital filters. In digital IIR filters, the output feedback is immediately apparent in the equations defining the output. Note that unlike FIR filters, in designing IIR filters it is necessary to carefully consider the "time zero" case in which the outputs of the filter have not yet been clearly defined.

Transfer Function Derivation

P is the feedforward filter orderbi are the feedforward filter coefficientsQ is the feedback filter orderai are the feedback filter coefficientsx[n] is the input signaly[n] is the output signal.When Rearranged

IIR Filter Z Transfer Function

Description Of Simple IIR Filter Block Diagram

StabilityThe transfer function allows us to judge whether or not a system is bounded-input, bounded-output (BIBO) stable. To be specific, the BIBO stability criteria requires that the ROC(Radius Of Convergence) of the system includes the unit circle. For example, for a causal system, all poles of the transfer function have to have an absolute value smaller than one. In other words, all poles must be located within a unit circle in the Z-plane.

The poles are defined as the values Z of which make the denominator of H(z) equal to 0 :

If aj =0 then the poles are not located at the origin of the

z-plane.

Applications of Digital filter

Communications SystemsAudio Systems Such As CD / DvdplayersInstrumentationImage Processing And EnhancementProcessing Of Seismic And Other

Geophysical SignalsProcessing Of Biological SignalsSpeech Synthesis

Advantages It has linear phase response.Thermal and environmental variation cannot

change the performance.It is possible to filter several input sequences

without any hardware replication.

DisadvantagesActually the speed operation totally depends on

the number of the arithmetic operation in the processor.

Finite word-length effect, which results quantizing noise and round-off noise.

It needs much longer time to design and develop the digital sequences though it can be used on other tasks or applications once developed.

ConclusionThe main utility of the analysis methods presented is in ascertaining how a given filter will affect the spectrum of a signal passing through it. Some of the concepts introduced were linearity, time-invariance, filter impulse response, difference equations, transient response, steady-state response, transfer functions, amplitude response, phase response, phase delay, group delay, linear phase, minimum phase, maximum phase, poles and zeros, filter stability, and the general use of complex numbers to represent signals, spectra, and filters.

Thank

You Under Guidance of Ms. Pradnya Vartak

Presented by Mr. Nilesh Rambahadur Jaiswar Mr.Kunal Dilip Vartak